Quasi-periodic flow

NEWHOUSE, J., PUELLE D., and TAKENS, F. "Occurence of strange axicm A attractors near quasi-periodic flows on t Ccrmun. Math. Phys. 1978, 64, 35-40. 

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

More recent experiments [62] concerning the viscous sublayer have shown a three-dimensional structure for turbulence near the wall. In a plane normal to the mean flow, counterrotating eddy pairs are involved (Fig. 6c), whereas in the direction of the mean flow, the motion is quasi-periodic (as described earlier). Since the wavelength along the mean flow is much larger than along the perimeter of the tube, a simplified bidimensional model may account only... 

As an example, if only quasi-steady flow elements are used with volume pressure elements, a model s smallest volume size (for equal flows) will define the timescale of interest. Thus, if the modeler inserts a volume pressure element that has a timescale of one second, the modeler is implying that events which happen on this timescale are important. A set of differential equations and their solution are considered stiff or rigid when the final approach to the steady-state solution is rapid, compared to the entire transient period. In part, numerical aspects of the model will determine this, but also the size of the perturbation will have a significant impact on the stiffness of the problem. It is well known that implicit numerical methods are better suited towards solving a stiff problem. (Note, however, that The Mathwork s software for real-time hardware applications, Real-Time Workshop , requires an explicit method presumably in order to better guarantee consistent solution times.)... 

The term melt fracture has been applied from the outset [9,13] to refer to various types of visible extrudate distortion. The origin of sharkskin (often called surface melt fracture ) has been shown in Sect. 10 to be related to a local interfacial instability in the die exit region. The alternating quasi-periodic, sometimes bamboo-like, extrudate distortion associated with the flow oscillation is a result of oscillation in extrudate swell under controlled piston speed due to unstable boundary condition, as discussed in Sect. 8. A third type, spiral like, distortion is associated with an entry flow instability. The latter two kinds have often been referred to as gross melt fracture. It is clearly misleading and inaccurate to call these three major types of extrudate distortion melt fracture since they do not arise from a true melt fracture or bulk failure. Unfortunately, for historical reasons, this terminology will stay with us and be used interchangeably with the phase extrudate distortion. ... 

Figure 18. (a) A schematic composite surface of section for nomotating T-shaped Hel2. (b) Idealization of surface of section indicating flow out of various phase-space regions. The hatched areas represent regions of quasi-periodic motion. [From S. K. Gray, S. A. Rice, and M. J. Davis, J. Phys. Chem. 90, 3470 (1986).]... 

A simple kinetics model of the three-state mechanism that takes into account bottlenecks to intramolecular energy transfer can be developed by splitting the phase space region A into the quasi-periodic motion region A and the highly chaotic region A2, with A = A + A2. Such a kinetics model is presented in Fig. 22. Let be the rate constant for flow of phase space points from Ai into... 

Other quasi-periodic textures have been reported, including birefringent stripes parallel to the flow direction (Gleeson et al. 1992). A low-shear-rate stripe texture observed in PBG solutions can be attributed to roll cells (Larson and Mead 1993), similar to those... 

Under most operating conditions a RFR eventually converges to a symmetric single-period operation so that the concentrations and temperature profiles after one flow reversal are a mirror image of those after the previous flow reversal. However, a cooled RFR may attain, under certain conditions, states with more complex periodicity, that is, states with period w > 1, nonsymmetric states and even complex quasi-periodic and chaotic states. 

Figure 2a depicts the Poincare section of the continuous flow stirrer when St = 1/4ti, and Re = 0.1. The Poincare sections are obtained by numerically tracking four passive tracer particles initially located at (0.005, -0.5), (0.005, 0.0), (0.005, 0.5) and (0.005, 1.0) during 10 convective time-scales PI/Uhs)- a quasi-periodic motion of the passive tracer particle that is initially located at (0.005, 0.0) results in a regular formation separating the upper and lower halves of the Poincare section. A zoomed image showing this KAM boundary is presented in Fig. 2c. The passive tracer particles initially located at the upper and lower halves of the channel entry cannot pass this global barrier. In addition to this, there are two unstirred zones called void zones surrounded by well stirred zone (chaotic sea) at the bottom half of the Poincare section. A zoomed image of these two void zones can be seen in Fig. 2b.

Particles that move in a laminar flow field with velocity gradient y experience shear and normal stresses which vary along/across the surface and induce particle rotation and deformation. The rotation of spheres is stable with an angular velocity of CO = 1/2 X 7 (Jeffery 1922 Trevelyan and Mason 1951), whereas aspherical particles or agglomerates rotate in a quasi-periodic or even chaotic manner (Blaser... 

Fig. 15 Stress time series recorded after start-up of flow experiment at a fixed shear rate 7= 25 s for different temperatures (a) T = 31.5°C, (b) T = 28.8°C, (c) T = 27.2°C, (d) T = 26.5 °C, (e) T = 26 °C. The sample under scrutiny is made of hexadecyltrimethylammonium p-toluenesulfonate (CTAT) at 2 wt. % mixed with 100 mM NaCl in water. The time sequences are found to be (a) time-independent, (b) periodic, (c) quasi-periodic, (d) intermittent and (e) chaotic. Reprinted with permission from R. Ganapathy and A.K. Sood [207]...

Two different reactions have presently been studied in the Couette flow reactor, namely the variants of the Belousov-Zhabotinsky [27-30, 32] and chlorite-iodide [29-33] reactions. The BZ reaction has revealed a rich variety of steady, periodic, quasi-periodic, frequency-locked, period-doubled and chaotic spatio-temporal patterns [27, 28], well described in terms of the diffusive coupling of oscillating reactor cells, the frequency of which changes continuously along the Couette reactor as the result of the imposed spatial gradient of constraints. This experimental observation has been successfully simulated with a schematic model of the BZ kinetics [68] and the recorded bifurcation sequences of patterns resemble those obtained when coupling two nonlinear oscillators. 

In a short time period, the dynamic model shown in Equation (3.13.1.1) at quasi-steady-state condition, OTR to microbial cells would be equal to oxygen molar flow transfer to the liquid phase.4... 

We started by analyzing the dominant modes of oscillation showed by oxygen content, streamflow, and ENSO using the Maximum Entropy Method (MEM) (Table 1). It is remarkable that both the streamflow to the reservoir and the AF showed common oscillations with ENSO at fi-equencies between 0.016 and 0.035 cycles month These frequencies are very close to the two main periods of ENSO (the quasi-biennial and quasi-quadrennial periods) [56]. Although we do not have a mechanistic explanation for this teleconnection (in fact the extratropical influence of ENSO is a hot topic in climate research, Merkel and Latif [57]), it is certainly difficult to propose an alternative explanation for the oscillations in AF and stream-flow observed at these frequencies. 

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